Finite Math Examples

{log}_{3} (2 x - 3) = 2 {log}_{3} (3) + {log}_{3} (3 x - 2)

$\log_{3} (2 x - 3) = 2 \log_{3} (3) + \log_{3} (3 x - 2)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Logarithm base

3

$3$ of

3

$3$ is

1

$1$ .

{log}_{3} (2 x - 3) = 2 \cdot 1 + {log}_{3} (3 x - 2)

$\log_{3} (2 x - 3) = 2 \cdot 1 + \log_{3} (3 x - 2)$

Step 1.1.2

Multiply

2

$2$ by

1

$1$ .

{log}_{3} (2 x - 3) = 2 + {log}_{3} (3 x - 2)

$\log_{3} (2 x - 3) = 2 + \log_{3} (3 x - 2)$

{log}_{3} (2 x - 3) = 2 + {log}_{3} (3 x - 2)

$\log_{3} (2 x - 3) = 2 + \log_{3} (3 x - 2)$

{log}_{3} (2 x - 3) = 2 + {log}_{3} (3 x - 2)

$\log_{3} (2 x - 3) = 2 + \log_{3} (3 x - 2)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

{log}_{3} (2 x - 3) - {log}_{3} (3 x - 2) = 2

$\log_{3} (2 x - 3) - \log_{3} (3 x - 2) = 2$

Step 3

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

{log}_{3} (\frac{2 x - 3}{3 x - 2}) = 2

$\log_{3} (\frac{2 x - 3}{3 x - 2}) = 2$

Step 4

Rewrite

{log}_{3} (\frac{2 x - 3}{3 x - 2}) = 2

$\log_{3} (\frac{2 x - 3}{3 x - 2}) = 2$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$

\neq

$\neq$

1

$1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

3^{2} = \frac{2 x - 3}{3 x - 2}

$3^{2} = \frac{2 x - 3}{3 x - 2}$

Step 5

Cross multiply to remove the fraction.

2 x - 3 = 3^{2} (3 x - 2)

$2 x - 3 = 3^{2} (3 x - 2)$

Step 6

Simplify

3^{2} (3 x - 2)

$3^{2} (3 x - 2)$ .

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Step 6.1

Raise

3

$3$ to the power of

2

$2$ .

2 x - 3 = 9 (3 x - 2)

$2 x - 3 = 9 (3 x - 2)$

Step 6.2

Apply the distributive property.

2 x - 3 = 9 (3 x) + 9 \cdot - 2

$2 x - 3 = 9 (3 x) + 9 \cdot - 2$

Step 6.3

Multiply.

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Step 6.3.1

Multiply

3

$3$ by

9

$9$ .

2 x - 3 = 27 x + 9 \cdot - 2

$2 x - 3 = 27 x + 9 \cdot - 2$

Step 6.3.2

Multiply

9

$9$ by

- 2

$- 2$ .

2 x - 3 = 27 x - 18

$2 x - 3 = 27 x - 18$

2 x - 3 = 27 x - 18

$2 x - 3 = 27 x - 18$

2 x - 3 = 27 x - 18

$2 x - 3 = 27 x - 18$

Step 7

Move all terms containing

x

$x$ to the left side of the equation.

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Step 7.1

Subtract

27 x

$27 x$ from both sides of the equation.

2 x - 3 - 27 x = - 18

$2 x - 3 - 27 x = - 18$

Step 7.2

Subtract

27 x

$27 x$ from

2 x

$2 x$ .

- 25 x - 3 = - 18

$- 25 x - 3 = - 18$

- 25 x - 3 = - 18

$- 25 x - 3 = - 18$

Step 8

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 8.1

Add

3

$3$ to both sides of the equation.

- 25 x = - 18 + 3

$- 25 x = - 18 + 3$

Step 8.2

Add

- 18

$- 18$ and

3

$3$ .

- 25 x = - 15

$- 25 x = - 15$

- 25 x = - 15

$- 25 x = - 15$

Step 9

Divide each term in

- 25 x = - 15

$- 25 x = - 15$ by

- 25

$- 25$ and simplify.

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Step 9.1

Divide each term in

- 25 x = - 15

$- 25 x = - 15$ by

- 25

$- 25$ .

\frac{- 25 x}{- 25} = \frac{- 15}{- 25}

$\frac{- 25 x}{- 25} = \frac{- 15}{- 25}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of

- 25

$- 25$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 25 x}{- 25} = \frac{- 15}{- 25}$

Step 9.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 15}{- 25}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Cancel the common factor of

$- 15$ and

$- 25$ .

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Step 9.3.1.1

Factor

$- 5$ out of

$- 15$ .

$x = \frac{- 5 (3)}{- 25}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor

$- 5$ out of

$- 25$ .

$x = \frac{- 5 \cdot 3}{- 5 \cdot 5}$

Step 9.3.1.2.2

Cancel the common factor.

$x = \frac{- 5 \cdot 3}{- 5 \cdot 5}$

Step 9.3.1.2.3

Rewrite the expression.

$x = \frac{3}{5}$

Step 10

Exclude the solutions that do not make

$\log_{3} (2 x - 3) = 2 \log_{3} (3) + \log_{3} (3 x - 2)$ true.

$No solution$